A principle of similarity for nonlinear vibration absorbers

With continual interest in expanding the performance envelope of engineering systems, nonlinear components are increasingly utilized in real-world applications. This causes the failure of wellestablished techniques to mitigate resonant vibrations. In particular, this holds for the linear tuned vibration absorber (LTVA), which requires an accurate tuning of its natural frequency to the resonant vibration frequency of interest. This is why the nonlinear tuned vibration absorber (NLTVA), the nonlinear counterpart of the LTVA, has been recently developed. An unconventional aspect of this absorber is that its restoring force is tailored according to the nonlinear restoring force of the primary system. This allows the NLTVA to extend the so-called Den Hartog’s equal-peak rule to the nonlinear range.
In this work, a fully analytical procedure, exploiting harmonic balance and perturbation techniques, is developed to define the optimal value of the nonlinear terms of the NLTVA. The developments are such that they can deal with any polynomial nonlinearity in the host structure. Another interesting feature
of the NLTVA, discussed in the paper, is that nonlinear terms of different orders do not interact with each other in first approximation, thus they can be treated separately. Numerical results obtained through the shooting method coupled with pseudoarclength continuation validate the analytical developments.